Rayleigh–Ritz method

In mathematics, cellular homology in algebraic topology is a homology theory for CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.

Definition

If X is a CW-complex with n-skeleton X_n, the cellular homology modules are defined as the homology groups of the cellular chain complex

$\cdots \to H_{n+1}(X_{n+1},X_{n})\to H_{n}(X_{n},X_{n-1})\to H_{n-1}(X_{n-1},X_{n-2})\to \cdots .$

[$X_{-1}$ is the empty set]

The group

$H_{n}(X_{n},X_{n-1})\,$

is free, with generators which can be identified with the n-cells of X. Let $e_{n}^{\alpha }$ be an n-cell of X, let $\chi _{n}^{\alpha }:\partial e_{n}^{\alpha }\cong S^{n-1}\to X_{n-1}$ be the attaching map, and consider the composite maps

$\chi _{n}^{\alpha \beta }:S^{n-1}\to X_{n-1}\to X_{n-1}/(X_{n-1}-e_{n-1}^{\beta })\cong S^{n-1}$

where $e_{n-1}^{\beta }$ is an $(n-1)$-cell of X and the second map is the quotient map identifying $(X_{n-1}-e_{n-1}^{\beta })$ to a point.

The boundary map

$d_{n}:H_{n}(X_{n},X_{n-1})\to H_{n-1}(X_{n-1},X_{n-2})\,$

is then given by the formula

$d_{n}(e_{n}^{\alpha })=\sum _{\beta }\deg(\chi _{n}^{\alpha \beta })e_{n-1}^{\beta }\,$

where $deg(\chi _{n}^{\alpha \beta })$ is the degree of $\chi _{n}^{\alpha \beta }$ and the sum is taken over all $(n-1)$-cells of X, considered as generators of $H_{n-1}(X_{n-1},X_{n-2})\,$.

Other properties

One sees from the cellular chain complex that the n-skeleton determines all lower-dimensional homology:

$H_{k}(X)\cong H_{k}(X_{n})$

for k < n.

An important consequence of the cellular perspective is that if a CW-complex has no cells in consecutive dimensions, all its homology modules are free. For example, complex projective space CPⁿ has a cell structure with one cell in each even dimension; it follows that for 0 ≤ k ≤ n,

$H_{2k}(\mathbb {CP} ^{n};\mathbb {Z} )\cong \mathbb {Z}$

and

$H_{2k+1}(\mathbb {CP} ^{n})=0.$

Generalization

The Atiyah-Hirzebruch spectral sequence is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary extraordinary (co)homology theory.

Euler characteristic

For a cellular complex X, let X_j be its j-th skeleton, and c_j be the number of j-cells, i.e. the rank of the free module H_j(X_j, X_j-1). The Euler characteristic of X is defined by

$\chi (X)=\sum _{0}^{n}(-1)^{j}c_{j}.$

The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of X,

$\chi (X)=\sum _{0}^{n}(-1)^{j}\;{\mbox{rank}}\;H_{j}(X).$

This can be justified as follows. Consider the long exact sequence of relative homology for the triple (X_n, X_{n - 1} , ∅):

$\cdots \to H_{i}(X_{n-1},\emptyset )\to H_{i}(X_{n},\emptyset )\to H_{i}(X_{n},X_{n-1})\to \cdots .$

Chasing exactness through the sequence gives

$\sum _{i=0}^{n}(-1)^{i}\;{\mbox{rank}}\;H_{i}(X_{n},\emptyset )=\sum _{i=0}^{n}(-1)^{i}\;{\mbox{rank}}\;H_{i}(X_{n},X_{n-1})\;+\;\sum _{i=0}^{n}(-1)^{i}\;{\mbox{rank}}\;H_{i}(X_{n-1},\emptyset ).$

The same calculation applies to the triple (X_{n - 1}, X_{n - 2}, ∅), etc. By induction,

$\sum _{i=0}^{n}(-1)^{i}\;{\mbox{rank}}\;H_{i}(X_{n},\emptyset )=\sum _{j=0}^{n}\;\sum _{i=0}^{j}(-1)^{i}\;{\mbox{rank}}\;H_{i}(X_{j},X_{j-1})=\sum _{j=0}^{n}(-1)^{j}c_{j}.$

References

• A. Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1.

• A. Hatcher: Algebraic Topology, Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the author's homepage.